4. (8 points total) Let G = (V. E) be a connected, edge-weighted, undirected graph with two distinguished vertices s and t, let T be a minimum-weight spanning tree of G, let P be the unique simple path in T between s and t, and let w be the weight of a maximum-weight edge on P. (a) (4 points) Prove that every path in G between s to t has at least one edge of weight greater than or equal to w. (b) (4 points) Let Q be a path between s and t with no edges of weight greater than w. (So, by part (a), Q has at least one weight-w edge.) Prove or disprove that the number of weight-w edges on Q is at least the number of weight-w edges on P. 4. (8 points total) Let G = (V. E) be a connected, edge-weighted, undirected graph with two distinguished vertices s and t, let T be a minimum-weight spanning tree of G, let P be the unique simple path in T between s and t, and let w be the weight of a maximum-weight edge on P. (a) (4 points) Prove that every path in G between s to t has at least one edge of weight greater than or equal to w. (b) (4 points) Let Q be a path between s and t with no edges of weight greater than w. (So, by part (a), Q has at least one weight-w edge.) Prove or disprove that the number of weight-w edges on Q is at least the number of weight-w edges on P